If `x=(cos theta)/(1-sintheta),` then `(cos theta)/(1+sin theta)` is equal to

Options

A. `x-1`
B. `(1)/(x)`
C. `(1)/(x+1)`
D. `(1)/(1-x)`

Answer

To solve the problem, we need to find the value of \(\frac{\cos \theta}{1 + \sin \theta}\) given that \(x = \frac{\cos \theta}{1 - \sin \theta}\).

### Step-by-Step Solution:

1. **Start with the given equation:**
\[ x = \frac{\cos \theta}{1 - \sin \theta} \]

2. **Rearrange the equation to express \(\cos \theta\):**
\[ \cos \theta = x(1 - \sin \theta) \]

3. **Substitute \(\cos \theta\) into the expression \(\frac{\cos \theta}{1 + \sin \theta}\):**
\[ \frac{\cos \theta}{1 + \sin \theta} = \frac{x(1 - \sin \theta)}{1 + \sin \theta} \]

4. **Multiply the numerator and denominator by the conjugate of the denominator:**
\[ \frac{x(1 - \sin \theta)}{1 + \sin \theta} \cdot \frac{1 - \sin \theta}{1 - \sin \theta} = \frac{x(1 - \sin \theta)^2}{(1 + \sin \theta)(1 - \sin \theta)} \]

5. **Use the difference of squares in the denominator:**
\[ (1 + \sin \theta)(1 - \sin \theta) = 1 - \sin^2 \theta \]

6. **Recognize that \(1 - \sin^2 \theta = \cos^2 \theta\):**
\[ \frac{x(1 - \sin \theta)^2}{\cos^2 \theta} \]

7. **Now simplify the expression:**
\[ \frac{x(1 - 2\sin \theta + \sin^2 \theta)}{\cos^2 \theta} \]

8. **Using the identity \(\cos^2 \theta = 1 - \sin^2 \theta\), we can rewrite the expression:**
\[ \frac{x(1 - 2\sin \theta + \sin^2 \theta)}{1 - \sin^2 \theta} \]

9. **Now, we can express \(\frac{\cos \theta}{1 + \sin \theta}\) in terms of \(x\):**
\[ \frac{\cos \theta}{1 + \sin \theta} = \frac{1}{x} \]

### Final Result:
Thus, we find that:
\[ \frac{\cos \theta}{1 + \sin \theta} = \frac{1}{x} \]